= Thomson principle
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For a finite connected undirected <electrical network> with positive <edge> <electrical conductances> $c_e$, the <Thomson principle> states
$$
R_{\mathrm{eff}}(a,z)=\min_{\theta}\sum_{e\in E}\frac{\theta(e)^2}{c_e},
$$
where $a\ne z$, each undirected <edge> is counted once with an arbitrary orientation, and $\theta$ ranges over <unit flows> with divergence $+1$ at $a$, $-1$ at $z$, and zero elsewhere. The expression is the dissipated energy for unit <electric current>.
To prove the minimum, let $i$ be the <current flow> arising from a <voltage> $u$ that drives one unit of <electric current>. Write another <unit flow> as $i+h$, so $h$ has zero divergence. Since $i(e)/c_e$ is the endpoint <voltage> difference, summation by parts gives
$$
\sum_e\frac{i(e)h(e)}{c_e}=\sum_xu(x)\operatorname{div}h(x)=0.
$$
Hence the energy of $i+h$ is the energy of $i$ plus the nonnegative energy of $h$. The energy of $i$ is $u(a)-u(z)=R_{\mathrm{eff}}(a,z)$ by <Ohm's law>. For two resistors of <electrical resistances> $2$ and $3$ in series, every <unit flow> uses both, giving <effective resistance> $5$.
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